Recent interrelated progress in Lorentzian, Finslerian and ......Lorentzian, Finslerian and Riemannian Geometry Miguel S anchez Caja Universidad de Granada Workshop on Lorentzian Geometry

1 Causal structure2 Visibility and lensing

3 Causal boundary

Recent interrelated progress inLorentzian, Finslerian and Riemannian Geometry

Miguel Sanchez Caja

Universidad de Granada

Workshop on Lorentzian GeometryLaboratoire Jacques-Louis Lions, Paris 6

November 14th, 2012

M. Sanchez Lorentz, Finsler and Riemann Geometry


3 Causal boundary

Introduction

Stationary to Randers correspondence. Equivalence:

Conformal structure of stationary spacetimes←→ Geometry of Randers spaces

Applicability:

→ Precise description of spacetime elements in terms ofFinsler counterparts

← New geometric elements and results in Randers spaces,extensible to general Finsler manifolds

Broad relation

Lorentzian Geometry ←→ Finsler Geometry

(including the Riemannian one!)



3 Causal boundary

Introduction



Applicability:



Broad relation





3 Causal boundary

Introduction



Applicability:



Broad relation





3 Causal boundary

Starting point

Normalized standard stationary spacetime:V = (R×M, gL = −1dt2 + π∗ω ⊗ dt + dt ⊗ π∗ω + π∗g)ω 1-form, g Riemannian metric on M, π : R×M → M projection

∂t timelike (future-directed) Killing vector fieldgL ≡ −dt2 + 2ωdt + g

Normalized: −1dt2. Useful for conformal elements such aslightlike vectors/geodesics (otherwise: −βdt2)

Global “standard” (but not unique) splitting not toorestrictive: it always hold locally and [Javaloyes & — ’08]:

A spacetime is (globally) conformal to a standard stationaryone iff it admits a complete timelike conformal vector fieldand it is distinguishing (and, so, strongly causal and causallycontinuous).



3 Causal boundary

Starting point

Normalized standard stationary spacetime:V = (R×M, gL = −1dt2 + π∗ω ⊗ dt + dt ⊗ π∗ω + π∗g)ω 1-form, g Riemannian metric on M, π : R×M → M projection

∂t timelike (future-directed) Killing vector fieldgL ≡ −dt2 + 2ωdt + g

Normalized: −1dt2. Useful for conformal elements such aslightlike vectors/geodesics (otherwise: −βdt2)

Global “standard” (but not unique) splitting not toorestrictive: it always hold locally and [Javaloyes & — ’08]:A spacetime is (globally) conformal to a standard stationaryone iff it admits a complete timelike conformal vector fieldand it is distinguishing (and, so, strongly causal and causallycontinuous).



3 Causal boundary

Starting point

Appearance of Finsler GeometryWith these elements ω, g construct the functions F± : TM → R

F±(v) =√

g(v , v) + ω(v)2 ± ω(v)

Finsler metrics of Randers type on M, “Fermat metrics”

Connection with the spacetime geometry (Cap, Jav. Mas. ’11):A curve γ(t) = (±t, c(t)), t ∈ [a, b] is lightlike and future/pastdirected iff F±(c) = 1. In this case:

the arrival time b− a is equal to the F±-length of c ,∫ ba F±(c)

(Fermat principle) γ is a pregeodesic iff c is a geodesic for F±

(i.e., a critical point of the arrival time/length functionalc 7→

∫F±(c) parametrized with F±-length.



3 Causal boundary

Starting point

Appearance of Finsler GeometryWith these elements ω, g construct the functions F± : TM → R

F±(v) =√

g(v , v) + ω(v)2 ± ω(v)

Finsler metrics of Randers type on M, “Fermat metrics”Connection with the spacetime geometry (Cap, Jav. Mas. ’11):A curve γ(t) = (±t, c(t)), t ∈ [a, b] is lightlike and future/pastdirected iff F±(c) = 1. In this case:

the arrival time b− a is equal to the F±-length of c ,∫ ba F±(c)

(Fermat principle) γ is a pregeodesic iff c is a geodesic for F±

(i.e., a critical point of the arrival time/length functionalc 7→

∫F±(c) parametrized with F±-length.



3 Causal boundary

Starting point

These elementary considerations suggest the possibility to relatethe conformal geometry of standard stationary spacetimes and thegeometry of the corresponding class of Finsler manifolds, i.e.Randers spaces.

Aims:

1 Causal structure ←→ Finslerian distances(Caponio, Javaloyes, — arxiv ’09)

2 Visibility and gravitational lensing ←→convexity of Finsler hypersurfaces(Caponio, Germinario, — arxiv ’11).

3 Causal boundaries ←→ Cauchy, Gromov and Busemannboundaries in Finslerian (and Riemannian) settings(Flores, Herrera, — ’10, ’12).



3 Causal boundary

Starting point

These elementary considerations suggest the possibility to relatethe conformal geometry of standard stationary spacetimes and thegeometry of the corresponding class of Finsler manifolds, i.e.Randers spaces. Aims:






3 Causal boundary

Starting point







3 Causal boundary

Starting point







3 Causal boundary

Preliminaries on Finsler distancesCausal HierarchyFurther results

FIRST PART

1. CAUSAL STRUCTURE



3 Causal boundary


Notion of Finsler and Randers metric

Let F : TM → R continuous and smooth away 0:

F Finsler metric: essentially, F a positively homogeneousnorm at each p ∈ M

Positive homogeneity F (λv) = λF (v) for λ > 0

The unit sphere is assumed strongly convex

Randers metric: R =√

g + ω2 + ω for some Riemanniang (and h := g + ω2) and 1-form ω.

In particular, Fermat metrics F± are Randers (and viceversa)

Reversed Finsler metric: F rev(v) := F (−v)

In particular,for Fermat metrics (F +)rev(v) = F−(v)

if ω 6= 0, Randers metrics are non-reversible (R 6= Rrev)



3 Causal boundary















3 Causal boundary















3 Causal boundary


Notion of generalized distance

Taking infimum of lengths connecting two points, each Finslermetric induces a generalized distance d . This means:

1 all the axioms of a distance hold but symmetry

2 for sequences xn: d(x , xn)→ 0 ⇐⇒ d(xn, x)→ 0

Centered at any point x0, there are forward balls (d(x0, x) < r)and backward balls (d(x , x0) < r) that may differ but generate thesame topology (in the Finslerian case, the manifold topology)

Symmetrized distance: ds(x , y) = (d(x , y) + d(y , x))/2

Remark Even in the Finslerian case, ds does not come from alength space.



3 Causal boundary











3 Causal boundary











3 Causal boundary











3 Causal boundary


Finslerian Hopf-Rinow

Theorem

Let (M,F ) be a Finsler manifold with generalized distance d.They are equivalent:

(a) d is forward (resp. backward) complete (Cauchy sequences).

(b) The Finsler manifold (M,F ) is forward (resp. backward)geodesically complete.

(c) At some (and then all) point p ∈ M, expp (resp. ˜expp) isdefined on all of TpM.

(d) Heine-Borel property: every closed and forward (resp.backward) bounded subset of (M, d) is compact.

Moreover, in this case (M,F ) is convex, i.e., every pair of pointsp, q ∈ M can be joined by a minimizing geodesic from p to q.



3 Causal boundary


Finslerian Hopf-Rinow

Remark. Relation with ds :

1 d is either forward or backward complete =⇒2 ds satisfies Heine-Borel ⇐⇒ all Bs(x , r) compact =⇒3 ds complete



3 Causal boundary


Basic idea

For p ∈M, d+ ≡ d distance of F+ ≡ F:

I +(0, p) determined by the graph of d+(p, ·):t0 × B+(p, t0) = I +(0, p) ∩ (t0 ×M)



3 Causal boundary


Results

Theorem

The slices t0 ×M are Cauchy hypersurfaces of (R×M, gL)iff d+ is forward and backward complete.

(R×M, g) is globally hyperbolic (causal + J+(z) ∩ J−(z ′)compact) iff B+

s (p, r) are compact ∀p ∈ M, r > 0(Heine-Borel property)

(R×M, gL) is causally simple (causal + J±(z) closed)iff (M,F ) is convex

(Full characterization of Causality, as standard stationary s-t arealways causally continuous.)



3 Causal boundary


Results

Theorem








3 Causal boundary


Results

Theorem








3 Causal boundary


Consequences for Finsler manifolds

Remark. Compactness of B+s (p, r) for a Randers metric

=⇒ (R×M, gL) is glob. hyp

=⇒ (R×M, gL) admits a spacelike Cauchy hypersurfaceMf = (f (x), x) : x ∈ M=⇒ The spacetime admits a splitting R×Mf with Cauchy slicesand Fermat metric Ff ≡ F − df=⇒ for some f , the F − df is forward and backward completeRanders metric! this property is extensible to any Finsler metric(Matveev’12).

In general, the compactness of B+s (p, r) (Heine-Borel) is

the optimal assumption for classical Finsler theorems(Myers, sphere...)



3 Causal boundary




=⇒ (R×M, gL) is glob. hyp=⇒ (R×M, gL) admits a spacelike Cauchy hypersurfaceMf = (f (x), x) : x ∈ M

=⇒ The spacetime admits a splitting R×Mf with Cauchy slicesand Fermat metric Ff ≡ F − df=⇒ for some f , the F − df is forward and backward completeRanders metric! this property is extensible to any Finsler metric(Matveev’12).





3 Causal boundary




=⇒ (R×M, gL) is glob. hyp=⇒ (R×M, gL) admits a spacelike Cauchy hypersurfaceMf = (f (x), x) : x ∈ M=⇒ The spacetime admits a splitting R×Mf with Cauchy slicesand Fermat metric Ff ≡ F − df

=⇒ for some f , the F − df is forward and backward completeRanders metric! this property is extensible to any Finsler metric(Matveev’12).





3 Causal boundary




=⇒ (R×M, gL) is glob. hyp=⇒ (R×M, gL) admits a spacelike Cauchy hypersurfaceMf = (f (x), x) : x ∈ M=⇒ The spacetime admits a splitting R×Mf with Cauchy slicesand Fermat metric Ff ≡ F − df=⇒ for some f , the F − df is forward and backward completeRanders metric! this property is extensible to any Finsler metric(Matveev’12).





3 Causal boundary




=⇒ (R×M, gL) is glob. hyp=⇒ (R×M, gL) admits a spacelike Cauchy hypersurfaceMf = (f (x), x) : x ∈ M=⇒ The spacetime admits a splitting R×Mf with Cauchy slicesand Fermat metric Ff ≡ F − df=⇒ for some f , the F − df is forward and backward completeRanders metric! this property is extensible to any Finsler metric(Matveev’12).





3 Causal boundary


Cauchy developments

A ⊂ V achronal setD+(A) = z ∈ V : γ ∩ A 6= ∅ for all γ past-inextensible causalcurve starting at pH+(A) = z ∈ D

+(A) : I +(z) ∩ D+(A) = ∅

Proposition

For A ⊂ M ≡ 0 ×M:D+(A) = (t, y) : 0 ≤ t < d+(x , y) ∀x 6∈ AH+(A) = (t, y) : t = infx 6∈Ad+(x , y)(= d+(M\A, y)

H+(A) is constructed from the level sets of d+(M\A, · )



3 Causal boundary


Cauchy developments

A ⊂ V achronal setD+(A) = z ∈ V : γ ∩ A 6= ∅ for all γ past-inextensible causalcurve starting at pH+(A) = z ∈ D

+(A) : I +(z) ∩ D+(A) = ∅

Proposition

For A ⊂ M ≡ 0 ×M:D+(A) = (t, y) : 0 ≤ t < d+(x , y) ∀x 6∈ AH+(A) = (t, y) : t = infx 6∈Ad+(x , y)(= d+(M\A, y)

H+(A) is constructed from the level sets of d+(M\A, · )



3 Causal boundary


Applications to Finsler

Remark. The results on horizons yield results on the distancefunction to a set in a Randers manifold, and viceversa.For example:

Theorem

Let C ⊂ M a closed subset, p ∈ M\C is a differentiable point ofthe distance from C iff p is crossed by exactly one minimizingsegment

generalizable to any Finsler manifold(Sabau-Tanaka’12)



3 Causal boundary


Applications to Finsler

Remark. The results on horizons yield results on the distancefunction to a set in a Randers manifold, and viceversa.For example:

Theorem

Let C ⊂ M a closed subset, p ∈ M\C is a differentiable point ofthe distance from C iff p is crossed by exactly one minimizingsegment generalizable to any Finsler manifold(Sabau-Tanaka’12)



3 Causal boundary

The physical problemPreliminaries on convex hypersurfacesMain ideas and results

SECOND PART

2. VISIBILITY AND LENSING



3 Causal boundary


Visibility of particles

Fermat principle (Kovner’90, Perlick’90): if a first arriving causalcurve connecting a point and a line exists then it is a lightlikegeodesic. BUT:

Does such a geodesic exist?

Will it remain in a (reasonably big) realistic region R× Dsuch that our conformally stationary model R×M?

Will massive particles (timelike geodesics) arrive in a finiteproper time (length) prior to their desintegration?

Lensing: can we receive more than once particles from thesame source emitted at the same or different times?

Problems related to convexity.



3 Causal boundary











3 Causal boundary











3 Causal boundary











3 Causal boundary











3 Causal boundary











3 Causal boundary


Previous: Riemannian convexity

(M, gR) Riemannian, D ⊂ M open domain with smooth ∂DNotions of convexity for ∂D:

1 Infinitesimal convexity: second fundamental form positivesemi-definite with respect to the inner normal⇐⇒ Hessφ negative semidefinite on T (∂D) for any smoothφ : D → [0,∞) with ∂D = φ−1(0) and φ regular on ∂D.

2 Local convexity:locally each exp(Tp(∂D) does not touch D.

Local =⇒ infinitesimal trivially (and pointwise)

Non-trivial converse (Do Carmo, Warner ’70)Bishop ’74 proved the converse but

1 his proof required smoothness C 4 (also for g)2 It cannot be extended to the Finslerian setting (Borisenko,

Olin ’10)



3 Causal boundary







Non-trivial converse (Do Carmo, Warner ’70)

Bishop ’74 proved the converse but1 his proof required smoothness C 4 (also for g)2 It cannot be extended to the Finslerian setting (Borisenko,

Olin ’10)



3 Causal boundary







Non-trivial converse (Do Carmo, Warner ’70)Bishop ’74 proved the converse but

1 his proof required smoothness C 4 (also for g)2 It cannot be extended to the Finslerian setting (Borisenko,

Olin ’10)



3 Causal boundary


General Finslerian results

Approach by Bartolo, Caponio, Germinario, — ’10:

previous notions extensible to Finslerian manifolds

intermediate notion: ∂D is geometrically convex when: nogeodesic in D connecting some p, q ∈ D touches ∂D.

Theorem

For any Finsler manifold, and domain D with C2,1

locboundary ∂D:

1 The infinitesimal, geometric and local notions of convexity areequivalent.



3 Causal boundary



Approach by Bartolo, Caponio, Germinario, — ’10:

previous notions extensible to Finslerian manifolds

intermediate notion: ∂D is geometrically convex when: nogeodesic in D connecting some p, q ∈ D touches ∂D.

Theorem


locboundary ∂D:




3 Causal boundary



Theorem (Bartolo, Caponio, Germinario, — ’10)


locboundary ∂D:


2 If all BDs (p, r) (in particular, if Bs(p, r)∩D) are compact: ∂D

is convex iff D is convex.

In the last case, if D is not contractible then any p, q ∈ D can beconnected by infinitely many geodesics contained in D.



3 Causal boundary



Theorem (Bartolo, Caponio, Germinario, — ’10)


locboundary ∂D:


2 If all BDs (p, r) (in particular, if Bs(p, r)∩D) are compact: ∂D

is convex iff D is convex.

In the last case, if D is not contractible then any p, q ∈ D can beconnected by infinitely many geodesics contained in D.



3 Causal boundary


Basic ideas: existence

Existence of connecting causal geodesics in a prescribed R× D

Lightlike geodesics. Optimal conditions:

1 (geometric) light-convexity of R× ∂D in R×M⇐⇒ Convexity of ∂D in (M,F ).

2 + completeness of the space of connecting curves:⇐⇒ Compactness of B D

s (p, r) (⇐ Bs(p, r) ∩ D)

Timelike geodesics. Reduction to the lightlike case:γ timelike geodesic in R×M with gL(γ′, γ′) = −c2

⇐⇒ γ(s) = (cs, γ(s)) is a lightlike geodesic in R× (R×M)with the product metric (≡ du2 + gL)



3 Causal boundary







s (p, r) (⇐ Bs(p, r) ∩ D)





3 Causal boundary







s (p, r) (⇐ Bs(p, r) ∩ D)





3 Causal boundary







s (p, r) (⇐ Bs(p, r) ∩ D)





3 Causal boundary


Basic ideas: multiplicity

Multiplicity (lensing)

Local, around a given lightlike pregeodesic γ(t) = (t, c(t)):existence of conjugate points for γ c admits conjugate point(gravitational lensing)

Global:non trivial topology of R× D D non-contractible(topological lensing)



3 Causal boundary


Basic ideas: multiplicity

Multiplicity (lensing)

Local, around a given lightlike pregeodesic γ(t) = (t, c(t)):existence of conjugate points for γ c admits conjugate point(gravitational lensing)

Global:non trivial topology of R× D D non-contractible(topological lensing)



3 Causal boundary


Results: lightlike geodesics

M ×R standard stationary, D ⊂ M a C 2 domain

Theorem

Assume that all BDs (p, r) are compact (which happens, in

particular, when Bs(p, r) ∩ D are compact ). They are equivalent:

1 (R× D, gL) is causally simple (⇔ (D,F ) is convex)

2 (∂D; F ) is convex

3 (R× ∂D; gL) is light-convex.

4 Any point w ∈ R× D and any line lq, q ∈ D connected inR× D by a future–pointing lightlike geodesic minimizing the(future) arrival time T (⇔ idem for past)

In this case, if D is not contractible, infinitely many connectinglightlike geodesics with diverging arrival times exist



3 Causal boundary




Theorem










3 Causal boundary




Theorem










3 Causal boundary




Theorem










3 Causal boundary




Theorem










3 Causal boundary




Theorem










3 Causal boundary


Results: timelike geodesics

R×M standard stationary, non-necessarily normalized (−βdt2),product metric for Ru ≡ (R, du2) and (R×M, gL), Πu projectionon the first factor, in addition to ΠM . Fermat Fβ on R×M:

Fβ =

√Π∗Mh +

Π∗udu2

β ΠM+ Π∗Mω =

√hβ + ω1.

(extra dimension plus non-conformal invariance)



3 Causal boundary


Results: timelike geodesics

Theorem


particular, when Bs(p, r) ∩ D are compact ). Then:

(Ru × ∂D; Fβ) is convex ⇔for any length l > 0, each point w ∈ R× D and line lq arejoined by a future (and a past) pointing timelike geodesic inR× D,with length l minimizing the arrival time (amongcausal curves of length l).

In this case, if D is not contractible, a sequence of such connectinggeodesics with diverging lengths exists.



3 Causal boundary


Further results: asympt. flat spacetimes

Asymptotically flat stationary spacetime:

(M, g) complete, outside a compact subset C , isdiffeomorphic to Rn \ B(0,R0), elements g , ω, β turningEuclidean with large radial coordinate r .

Model isolated systems -gravity outside a star.

Makes sense to speak on (stationary) large ballsB(0,R)×R ⊂ R×M and spheres.

Application: Large spheres in asympt. flat spacetimes are alwayslight-convex (and all the previous results are applicable) but,typically, (including reasonable matter, when gravity attracts) theyare not time-convex



3 Causal boundary


Further results: asympt. flat spacetimes

Asymptotically flat stationary spacetime:

(M, g) complete, outside a compact subset C , isdiffeomorphic to Rn \ B(0,R0), elements g , ω, β turningEuclidean with large radial coordinate r .

Model isolated systems -gravity outside a star.

Makes sense to speak on (stationary) large ballsB(0,R)×R ⊂ R×M and spheres.

Application: Large spheres in asympt. flat spacetimes are alwayslight-convex (and all the previous results are applicable) but,typically, (including reasonable matter, when gravity attracts) theyare not time-convex



3 Causal boundary

Riemannian boundariesFinslerian boundariesC-boundary on stationary s-t

THIRD PART

3. CAUSAL BOUNDARY



3 Causal boundary


Introduction

Causal boundary ∂V of a spacetime V :

Involved structure: conformal structure (Causality).Intrinsic alternative to common Penrose conformal boundaryapplicable to any strongly causal spacetime

Purpose: attach a boundary endpoint P ∈ ∂V to anyinextensible future or past directed timelike curve γ



3 Causal boundary


Introduction

Basic idea: the boundary point would be represented byP = I−(γ) or F = I +(γ) or, more precisely, a pair (P,F ).



3 Causal boundary


Introduction

Long story from Geroch, Kronheimer & Penrose ’72 until its recentredefinition Flores, Herrera & — ’11):

As a point set, the completion V is composed by pairs (P,F )(“(IP,IF)”, for example V 3 p ≡ (I +(p), I−(p)) ∈ V )

Chronological relation: (P,F ) (P ′,F ′)⇔ F ∩ P ′ 6= ∅



3 Causal boundary


Introduction

Long story until its recent redefinition:

As a point set, the completion V is composed by pairs (P,F )(“(IP,IF)”, for example V 3 p ≡ (I +(p), I−(p)) ∈ V )Chronological relation: (P,F ) (P ′,F ′)⇔ F ∩ P ′ 6= ∅Subtle topology... non always Hausdorff



3 Causal boundary


Introduction

When computed for standard stationary spacetimes, relations with

Cauchy boundary

Gromov boundary

Busemann-type boundary

for Randers manifolds:

previous study for any Riemannian and Finslerianmanifold with interest by itself



3 Causal boundary


Introduction

When computed for standard stationary spacetimes, relations with

Cauchy boundary

Gromov boundary

Busemann-type boundary

for Randers manifolds:

previous study for any Riemannian and Finslerianmanifold with interest by itself



3 Causal boundary


Cauchy boundary for Riemannian manifolds

Remark: it may be non-locally compact



3 Causal boundary


Gromov boundary for Riemannian manifolds

Classical Gromov’s compactification for complete Riemannianmanifolds

L1(M, g) 1-Lipschitz functions (pointwise topology)

x ∈ M can be seen in L1(M, g) as dx : y 7→ d(x , y) and alsodx + C for any C ∈ Rf ∼ f ′ ⇐⇒ f − f ′ =constant (quotient topology)

each x ∈ M is represented in L1(M, g)/ ∼ as the class of −dx

MG = closure of M in L1(M, g)/ ∼



3 Causal boundary



What about if (M, g) is not complete? Repeat construction:

Cauchy MC : completion, no compactification (MC may benon-locally compact)

Gromov MG : compactification even in the incomplete case.

1 MC → MG in a natural way and continuous but:the inclusion is an embedding ⇔ MC is locally compact

2 MG = M ∪ ∂CGM ∪ ∂GM

∂CGM: limits of bounded sequences (∂CM ⊂ ∂CGM)∂GM: limits of unbounded sequences



3 Causal boundary







2 MG = M ∪ ∂CGM ∪ ∂GM




3 Causal boundary







2 MG = M ∪ ∂CGM ∪ ∂GM




3 Causal boundary


Busemann boundary for Riemannian manifolds

M ≡ (M, g) connected Riemannian manfiold

Typically, Busemann functions are defined when c is a ray(half unit geodesic with no cut locus)bc(x0) = limt→∞(t − d(x0, c(t))) for all x0 ∈ M

Eberlein & O’Neill developed a compactification of anyHadamard manifold in terms of Busemann functions (conetopology), which coincides with Gromov’s one



3 Causal boundary




Typically, Busemann functions are defined when c is a ray(half unit geodesic with no cut locus)bc(x0) = limt→∞(t − d(x0, c(t))) for all x0 ∈ M




3 Causal boundary




Typically, Busemann functions are defined when c is a ray(half geodesic with no cut locus)bc(x0) = limt→∞(t − d(x0, c(t))) for all x0 ∈ M


We will admit a “Busemann” function for any curvec : [0,Ω)→ M with |c| ≤ 1bc(x0) = limt→Ω(t − d(x0, c(t)))

bc is ∞ at some x0 ∈ M iff bc ≡ ∞.B(M): set of finite Busemann functions



3 Causal boundary



As a subset B(M) ⊂ L1(M, g)Busemann completion MB = B(M)/ ∼ (quotient by additiveconstant –included as a subset in MG )

B(M), and then MB , are topological spaces with thechronological topology (different to the induced fromL1(M, g)) defined by means of a limit operator Lgiven fn ⊂ B(M), the subset L(fn) ⊂ B(M) is defined by:f ∈ L(fn) iff

(a) f ≤ lim infn fn and(b) ∀g ∈ B(M) with f ≤ g ≤ lim supn fn, it is g = f .



3 Causal boundary




B(M), and then MB , are topological spaces with thechronological topology (different to the induced fromL1(M, g))

defined by means of a limit operator Lgiven fn ⊂ B(M), the subset L(fn) ⊂ B(M) is defined by:f ∈ L(fn) iff




3 Causal boundary




B(M), and then MB , are topological spaces with thechronological topology (different to the induced fromL1(M, g)) defined by means of a limit operator Lgiven fn ⊂ B(M), the subset L(fn) ⊂ B(M) is defined by:f ∈ L(fn) iff




3 Causal boundary



Properties of the Busemann completion:

1 MB is sequentially compact

2 MB is T1, and points in ∂BM may be non-T2 related

3 MC → MB → MG (naturally) but MB topology is coarser.

4 MB = MG (as pointsets and, then, topologically)⇐⇒ ∂BM is Hausdorff

Discrepancy: MB compactifies directions (finite or asympt)MG may contain non-endpoints of curves in M

5 MB = M ∪ ∂CM (finite directions) ∪∂BM (asymptotic)

MC : Busemann functions for curves c with Ω <∞MB: Busemann for c with Ω =∞



3 Causal boundary


Cauchy boundary for Finslerian manifolds

Cauchy completions for a Finsler manifold:(M, d), d associated to F

1 Two types of Cauchy sequences (ordering)Cauchy boundaries ∂+

C M, ∂−C M plus the symmetrized one∂sCM = ∂−C M ∩ ∂+

C M

2 The extension dQ of d to, say, M+C = M ∪ ∂+

C M is not ageneralized distance but a quasidistance( dQ(xn, x)→ 0 6=⇒ dQ(x , xn)→ 0)

Topology on M+C generated by the forward balls different to

generated by backward balls∂+C M may be only a T0 space



3 Causal boundary


Cauchy boundary for Finslerian manifolds

Cauchy completions for a Finsler manifold:(M, d), d associated to F

1 Two types of Cauchy sequences (ordering)Cauchy boundaries ∂+

C M, ∂−C M plus the symmetrized one∂sCM = ∂−C M ∩ ∂+

C M

2 The extension dQ of d to, say, M+C = M ∪ ∂+

C M is not ageneralized distance but a quasidistance( dQ(xn, x)→ 0 6=⇒ dQ(x , xn)→ 0)

Topology on M+C generated by the forward balls different to

generated by backward balls∂+C M may be only a T0 space



3 Causal boundary


Finslerian Gromov completions

Gromov completions for a Finsler manifold:

1 Non-symmetric notions of LipschitzianL+

1 (M, d): f (y)− f (x) ≤ d(x , y)(L−1 (M, d) : f (x)− f (y) ≤ d(x , y))

2 Two Gromov’s compactifications M±G ,say: M+

G = closure of M in L+1 (M, d)/ ∼

In a natural way, i : M+C → M+

G but:

i continuous iff the backward balls generate a finer topologyon M+

C than the forward balls

i embedding when:

M+C is locally compact [also Riemannian condition] AND

dQ is a generalized distance



3 Causal boundary









G but:



i embedding when:





3 Causal boundary









G but:



i embedding when:





3 Causal boundary


Finslerian Busemann completions

Busemann completions for a Finsler manifold:

1 There are also two completions M±B constructed by usingBusemann functions (which depends on the order of thearguments in d)

2 As in the Riemannian case, each M±B is naturally included inM±G (with a coarser topology) and both boundaries coincideiff ∂±B M is Hausdorff.

3 The spacetime viewpoint suggests further relations betweenM+

B and M−B .



3 Causal boundary





2 As in the Riemannian case, each M±B is naturally included inM±G (with a coarser topology) and

both boundaries coincideiff ∂±B M is Hausdorff.


B and M−B .



3 Causal boundary







B and M−B .



3 Causal boundary







B and M−B .



3 Causal boundary


Appearance of the Busemann functions

Standard stationary spacetime:(g Riemannian on M with distance d , π : R×M → M projection)

V = (R×M, gL = −dt2 + 2dt ⊗ π∗ω + π∗g)

Fermat metric(with associated generalized distance d+)

F +(v) =√

g(v , v) + ω(v)2 ∀v ∈ TM

Aim: computation of IP’s (and dual IF’s)P = I−[γ], future-directed timelike curve

γ(t) = (t, c(t)), t ∈ [α,Ω), |c | < 1



3 Causal boundary



Standard stationary spacetime:(g Riemannian on M with distance d , π : R×M → M projection)

V = (R×M, gL = −dt2 + 2dt ⊗ π∗ω + π∗g)

Fermat metric(with associated generalized distance d+)

F +(v) =√

g(v , v) + ω(v)2 ∀v ∈ TM

Aim: computation of IP’s (and dual IF’s)P = I−[γ], future-directed timelike curve

γ(t) = (t, c(t)), t ∈ [α,Ω), |c | < 1



3 Causal boundary



1 Characterization of (t0, x0) (t1, x1) ⇐⇒ d+(x0, x1) < t1 − t0

2 Application to P = I−[γ], γ(t) = (t, c(t)):P = (t0, x0) ∈ V : (t0, x0) γ(t) for some t ∈ [α,Ω)= (t0, x0) ∈ V : t0 < t − d+(x0, c(t)) for some t ∈ [α,Ω)= (t0, x0) ∈ V : t0 < limt→Ω(t − d+(x0, c(t)))= (t0, x0) ∈ V : t0 < b+

c (x0)

Busemann forward function:b+c (x0) = limt→Ω(t − d+(x0, c(t)))



3 Causal boundary



1 Characterization of (t0, x0) (t1, x1) ⇐⇒ d+(x0, x1) < t1 − t0

2 Application to P = I−[γ], γ(t) = (t, c(t)):P = (t0, x0) ∈ V : (t0, x0) γ(t) for some t ∈ [α,Ω)= (t0, x0) ∈ V : t0 < t − d+(x0, c(t)) for some t ∈ [α,Ω)= (t0, x0) ∈ V : t0 < limt→Ω(t − d+(x0, c(t)))= (t0, x0) ∈ V : t0 < b+

c (x0)

Busemann forward function:b+c (x0) = limt→Ω(t − d+(x0, c(t)))


Fundamental correspondence1 Causal structure2 Visibility and lensing

3 Causal boundary


Fundamental correspondence

B+(M): set of all Busemann functions b+c for (M,F +)

IP‘s on V ≡ B+(M) ∪bc ≡ ∞

1 Past of points (PIP’s): converging c (⇒ Ω <∞)

2 Past of inextensible curves (TIP’s): non-converging c(⇐ Ω =∞)

Remark: for the causal boundary no quotient in the set ofBusemann functions must be carried out



3 Causal boundary


Computation: c-boundary for static spacetimes

Parts of ∂V

i+, i− apexes (bc ≡ ∞)of a “double cone ”

Horismotic (as lightlikewith no cut points) lineson ∂BM starting at i±

Timelike lines on ∂cM(unique non-trivialS-pairs) connecting i+, i−

Recall: with the chr-topology



3 Causal boundary



Parts of ∂V







3 Causal boundary



Parts of ∂V







3 Causal boundary


Computation: c-boundary for stationary spacetimes

Parts of ∂V

“Static part”:i+, i− “apexes” of twodistinct conesHorismotic lines on ∂±BMTimelike lines on ∂sCM(composed of S-pairs)

Locally horismotic lineson ∂±C M\∂sCMThe lines arrive both i+

and i− iff pairings withlines on ∂∓C M\∂sCM



3 Causal boundary


Computation: c-boundary for stationary spacetimes

Parts of ∂V

“Static part”:i+, i− “apexes” of twodistinct conesHorismotic lines on ∂±BMTimelike lines on ∂sCM(composed of S-pairs)

Locally horismotic lineson ∂±C M\∂sCMThe lines arrive both i+

and i− iff pairings withlines on ∂∓C M\∂sCM


Documents

Recent interrelated progress in Lorentzian, Finslerian and ......Lorentzian, Finslerian and Riemannian Geometry Miguel S anchez Caja Universidad de Granada Workshop on Lorentzian Geometry